- Probability
- Probability model
- Statistical model
- Differences between a statistical model and a probability model
A probability model is a mathematical representation of a random phenomenon. It is defined by its sample space,
events within the sample space, and probabilities associated with each event.
- The sample space S for a probability model is the set of all possible outcomes.
- An event A is a subset of the sample space S.
- A probability is a numerical value assigned to a given event A. The probability of an event is written P(A), and describes the long-run relative frequency of the event.
The first two basic rules of probability are the following:
- Any probability P(A) is a number between 0 and 1 (0 < P(A) < 1).
- The probability of the sample space S is equal to 1 (P(S) = 1).
A statistical model is a special class of mathematical model. What distinguishes a statistical model from other
mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via
mathematical equations, some of the variables do not have specific values, but instead have probability distributions;
i.e. some of the variables are stochastic. In the example above, ε is a stochastic variable; without that variable, the
model would be deterministic.
Statistical models are often used even when the physical process being modeled is deterministic. For instance, coin
tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process).
There are three purposes for a statistical model, according to Konishi & Kitagawa.[4]
- Predictions
- Extraction of information
- Description of stochastic structures
The main difference is that a probability model is only one (known) distribution, while a statistical model is a set of
probability models; the data is used to select a model from this set or a smaller subset of models that better
(in a certain sense) describe the phenomenon (in the light of the data).
